I am studying the equations that Fresnel used/invented in his work on lenses. I'm interested in

(1) a simple explanation of the equations and

(2) how the equations transferred into actual lenses.

(3) Then, for the magazine that ALF is putting out for children in grades 5 and 6, I'm looking for photos taken by members of each order of lens. They will be published with full credit given to the photographer. We are going to have a two-four page spread on Fresnel lenses. The first part will be a matching contest to have the students try to match the order of the light in the individual pictures to the name of the order (first, second...etc). The next part of the spread will be the actual answers and a brief explanation of each type of lens.

Not an easy subject or easy mathematics to understand or explain. The best help to understand the various equations and how they applied to Fresnel lens design and their affect on the light itself is to obtain the video or DVD titled "Lighthouse Illumination Technical Edition" from Artworks Florida. This video was created by Dan Spinella and is an absolutely great technical reference. I think the Video and DVD are available through Lighthouse Depot.

I do not believe the Fresnel equations were the primary considerations that influenced the specific design of the Fresnel lens. Rather, I believe it was practical knowledge acquired when developing and testing these theoretical equations.

(1) a simple explanation of the equations

Augustin Fresnel developed mathematical equations in an effort to prove a theory, a theory widely discounted by the traditional physicists at the time. This was the Wave Theory that describes light as waves that spread out from the source and that each color has its own wave length. The majority, the traditionalists supported the Corpuscular Theory, a theory that light was comprised of rapid emissions of tiny particles or energy packets.

In the field of physics, mathematical equations are used to describe behavior of matter and energy. For these equations to be valid and accepted they must describe all behaviors of such matter and energy under any and all conditions.

Fresnel first developed equations to describe the behavior of dispersed light. He then went on to expand these to include reflected, refracted, double refracted and polarized light. In this work he used mirrors, lenses, prism and other objects.

When he published his first work the Corpuscular crowd attempted to use his equations to disprove the Wave Theory. They did this by bench testing using optical devices available at the time. Low and behold they succeeded, succeeded in proving he was right. The tests proved the Wave Theory was correct and that the Corpuscular Theory had big holes in it. One big hole was the Corpuscular Theory could not explain the rainbow effect, which was easily explained by Wave Theory.

This proof of Wave Theory brought about a major paradigm shift in the world of optical physics. It has been said that Fresnel’s work on optical effects caused by the motion of objects was significant in Einstein’s theories of relativity.

The reflection, refraction and transmission characteristics of glass prisms were known before Fresnel first entered the optical physics field. What Fresnel did was to apply prism “technology” to existing convex lens design. This innovative application of prism optics would not have required a great reliance on his previously developed theoretical equations.

The challenge with the lens design was to obtain the correct physical and optical configuration, the placement of the grooves to achieve the optimum prism effect to obtain maximum light transmission and aligned precisely to produce a focused and concentrated beam of light. This would not have required application of the complicated Wave Theory equations. The math for this would involve calculations of angles and distances relative to the focal point of the lens.

Fresnel’s first design was a bull’s eye lens with annular rings surrounding the central convex lens. His next design was the drum lens or cylinder lens. These were dioptric lenses that focus light by refracting or bending. He then devised a system using mirrors to capture the light escaping above and below the lens. This did not perform to his liking because of light loss in reflection from silvered mirrors. Shortly before his death in 1827 he replaced the mirrors with reflecting prisms mounted above and below the Fresnel lens. These prisms served to focus light by both refracting and reflecting light from the illumination source. This final design by Fresnel is the catadioptric configuration of the 1st through 6th Order lenses.

I would suggest that it was Fresnel’s prior hands-on work with lenses, mirrors and prisms and his knowledge of what could be done with these that inspired his thinking. This led to the creation of first Fresnel lens when he was employed by the French Lighthouse Commission in 1822.

As he demonstrated in his earlier the wave theory work, Fresnel was capable of thinking outside the box. He was not constrained by the pre-supposed limits of dioptric lens technology of the time. Unlike many theorists, Fresnel excelled at practical application. He was not a theoretical physicist. He was an experienced civil engineer.

Fresnel was also working in relative isolation without realizing that some of his theories were already out there. He succeeded in proving them to be correct. However, there had to be a time when his lighthouse work made him realize that Fresnel lenses would refocus points of light to be a beam of light that can travel over distance. I would imagine that there was math behind this, and this is what I'm attempting to study. I've looked into and studied refraction and many of the subtopics of wave theory and have read some of the books in college libraries on wave theory. There seems to be one book by or about Fresnel that I can't locate, and it is in French, so it won't do me much good unless it has been translated.

Fresnel was an interesting personality. He had the personality type that made him want to make a lasting impression upon the educated world and into the future that he made a difference. He never really felt that that happened during his life (according to many sources) since he seemed to be resubstantiating the work of others at many points in time. As Dave pointed out, his success is in the practical applications of wave theory and showing that it was a theory that had practicality. As we all know today, Fresnel type lenses are wide-spread in use (such as car tail lights and traffic lights).

In November 1864, the book “Memoire sur l'eclairage et le balisage des cotes de France” was written by Léonce Reynaud, who served as Inspector General of Bridges and Roads, Director of the Lighting and Buoy Service for the French Government.

Translating as “Memoir on the light-house illumination of the coasts of France,” the book recounted the history of French lighthouse illumination, focusing in great depth on studies conducted on various illumination systems, fuels, visibility ranges of different colored lights, and the effects of atmospheric conditions on same. Perhaps most importantly, Reynaud paid great attention to Augustin Fresnel’s lenticular apparatus, including explanations of much of the physics involved in the lenticular system.

The book was considered such a valuable treatise on lighthouse illumination that it was translated into English for the US Lighthouse Board in 1871 by Rear Admiral Thornton A. Jenkins, Naval Secretary with the Board.

While copies of the book show up in the online book stores from time to time for $450.00 and up, we are fortunate that the entire 1871 version has been digitized and placed online as part of the “Making Of America” collection.

Peter.
I found this concerning calculations, which is a reference to material in the 1911 Encyclopedia Britanica, Volume V16, Page 649.

Optical Calculations.
The mathematical theory of optical apparatus for lighthouses and formulae for the calculations of profiles will be found in the works of the Stevensons, Chance, Allard, Reynaud, Ribiere and others. Particulars of typical lighthouse apparatus will be found in tables VI. and VII. 4.

And to Terry,
I broused through some of web pages of the translated Reynaud document and there discovered that some of the calibration work was done in the large wine cellars of Paris. Hmmmm. Kind of makes one wonder about the accuracy and precision of the testing.

Terry and Dave, thanks for the helpful information. Terry, thanks for the reference url to the book. I keep seeing the French version out there but could never verify that there was an English version. $450 is steep to get a personal copy, but reading on-line is fine too.

I have not had a problem like this and with a slow speed connection. If you can resolve your problem, copy and print the index on pages 141-144 to use as a guide to find specific topics in the many individual pages.

you can also buy a replica of the web site production for something like $13.00 which is reasonable. I can't print it at work so I'll play on the modem tonight at home and see if I can print pages and get the missing pages.